Linear Lines

Question

A linear line is defined as ax-3by=-66, and it passes through the points (-6,4) and (12,25), Determine the values of a and b.

Answer 1

I think you meant to say "The equation of a line is given by ...."  It is redundant to call something a "linear line".  Anyway,

 

y = mx + y₀

 

m = (25 - 4)/[12 - (-6)] = 21/18 = 7/6

 

y = (7/6)x + y₀

 

25 = (7/6)(12) + y₀

 

y₀ = 11

 

y = (7/6)x + 11

 

Multiply both sides by 6.

 

6y = 7x + 66

 

7x - 6y = -66

 

a = 7

 

-3b = -6

 

b = 2

 

Test the answer against the 2 given points.

 

Answer 2

(-6,4) ---> x=-6 and y=4

 

(-6)a - 3b(4) = -66

 -6a - 12b = -66

  -6(a+2b) = -66

    a+2b = 11

      a = 11 - 2b

 

 

 

(12,25) ---> x = 12 and y = 25

 

 12a - 3b(25) = -66

  12a - 75b = -66

 

 12(11-2b) - 75b = -66

 132 - 24b - 75b  = -66

  132 - 99b = -66

 

   -99b = -66 - 132

   -99b = -198

   b = 2 

  a = 11 - 2b = 11 - 2(2) = 11 - 4 = 7

 

 

So the line is ax - 3by = -66

                     7x - 3(2)y = -66

                      7x - 6y = -66